A general procedure is provided for comparing correlation coefficients between optimal linear composites. The procedure allows computationally efficient significance tests on independent or dependent multiple correlations, partial correlations, and canonical correlations, with or without the assumption of multivariate normality. Evidence from some Monte Carlo studies on the effectiveness of the methods is also provided.

A family of solutions for linear relations among k sets of variables is proposed. It is shown how these solutions apply for k = 2, and how they can be generalized from there to k ≥ 3.

The family of solutions depends on three independent choices: (i) to what extent a solution may be influenced by differences in variances of components within each set; (ii) to what extent the sets may be differentially weighted with respect to their contribution to the solution—including orthogonality constraints; (iii) whether or not individual sets of variables may be replaced by an orthogonal and unit normalized basis.

Solutions are compared with respect to their optimality properties. For each solution the appropriate stationary equations are given. For one example it is shown how the determinantal equation of the stationary equations can be interpreted.

In this paper we study the interrelationships between two sets of data measured on the same subjects via redundancy analysis. We consider redundancy analysis from an inferential point of view. Under the hypothesis of multinormality, tests of significance are obtained for each successive redundancy component so that only the significant factors are retained for prediction purposes. An example illustrates the method.

Canonical redundancy analysis provides an estimate of the amount of shared variance between two sets of variables and provides an alternative to canonical correlation. The proof that the total redundancy is equal to the average squared multiple correlation coefficient obtained by regressing each variable in the criterion set on all variables in the predictor set is generalized to the case in which there are a larger number of criterion than predictor variables. It is then shown that the redundancy for the criterion set of variables is invariant under affine transformation of the predictor variables, but not invariant under transformation of the criterion variables.

This paper explores some of the problems associated with traditional canonical correlation. A response surface methodology is developed to examine the stability of the derived linear functions, where one wishes to investigate how much the coefficients can change and still be in an ɛ-neighborhood of the globally optimum canonical correlation value. In addition, a discrete (or constrained) canonical correlation method is formulated where the derived coefficients of these linear functions are constrained to be in some small set, e.g., {1, 0, −1}, to aid in the interpretation of the results. An example concerning the psychographic responses of Wharton MBA students of the University of Pennsylvania regarding driving preferences and life-style considerations is provided.

The Maxbet method is an alternative to the method of generalized canonical correlation analysis and of Procrustes analysis. Contrary to these methods, it does not maximize the inner products (covariances) between linear composites, but also takes their sums of squares (variances) into account. It is well-known that the Maxbet algorithm, which has been proven to converge monotonically, may converge to local maxima. The present paper discusses an eigenvalue criterion which is sufficient, but not necessary for global optimality. However, in two special cases, the eigenvalue criterion is shown to be necessary and sufficient for global optimality. The first case is when there are only two data sets involved; the second case is when the inner products between all variables involved are positive, regardless of the number of data sets.

The name “Roy's largest root” and similar names are used in practice to label two different but functionally related statistics—one proportional to anF, and the other, a squared canonical correlation. This note presents the logic that leads to the two formulations, states which statistic some popular statistical packages use, and shows the possible source of this inconsistency in the original work of Roy (1953) and Heck (1960).

When measuring the same variables on different “occasions”, two procedures for canonical analysis with stationary compositing weights are developed. The first, SUMCOV, maximizes the sum of the covariances of the canonical variates subject to norming constraints. The second, COLLIN, maximizes the largest root of the covariances of the canonical variates subject to norming constraints. A characterization theorem establishes a model building approach. Both methods are extended to allow for Cohort Sequential Designs. Finally a numerical illustration utilizing Nesselroade and Baltes data is presented.

The interrelationships between two sets of measurements made on the same subjects can be studied by canonical correlation. Originally developed by Hotelling [1936], the canonical correlation is the maximum correlation between linear functions (canonical factors) of the two sets of variables. An alternative statistic to investigate the interrelationships between two sets of variables is the redundancy measure, developed by Stewart and Love [1968]. Van Den Wollenberg [1977] has developed a method of extracting factors which maximize redundancy, as opposed to canonical correlation.

A component method is presented which maximizes user specified convex combinations of canonical correlation and the two nonsymmetric redundancy measures presented by Stewart and Love. Monte Carlo work comparing canonical correlation analysis, redundancy analysis, and various canonical/redundancy factoring analyses on the Van Den Wollenberg data is presented. An empirical example is also provided.

The perturbation theory of the generalized eigenproblem is used to derive influence functions of each squared canonical correlation coefficient and the corresponding canonical vector pair. Three sample versions of these functions are described and some properties are noted. As particular applications, the influence function of the squared multiple correlation coefficient and influence functions of eigenvalues and eigenvectors in correspondence analysis are obtained. Three numerical examples are briefly discussed.

A distinction is drawn between redundancy measurement and the measurement of multivariate association for two sets of variables. Several measures of multivariate association between two sets of variables are examined. It is shown that all of these measures are generalizations of the (univariate) squared-multiple correlation; all are functions of the canonical correlations, and all are invariant under linear transformations of the original sets of variables. It is further shown that the measures can be considered to be symmetric and are strictly ordered for any two sets of observed variables. It is suggested that measures of multivariate relationship may be used to generalize the concept of test reliability to the case of vector random variables.

The purpose of this study was to examine the relationship of eight categories of leisure participation and enjoyment with ten selected demographic variables. A Likert-type questionnaire was designed to ascertain the degree of participation and enjoyment in 89 activities. The activities were grouped in the following leisure categories: aesthetic, civic, intellectual, mass-media, physical, social, spiritual and touristic.

The sample consisted of three executive board members and five non-board members for each of the 229 senior citizen clubs currently affiliated with the New Brunswick Senior Citizen's Federation. Canonical correlation analyses were used to investigate the relationship stated in the two research questions. Three meaningful canonical variates were respectively identified for participation and enjoyment. On the basis of the results of the study, there seems to be some justification to take into account the particular ethnic orientation of senior clubs in the delivery of the various leisure programs.

Johnstone Strait provides important summer habitat for the northern resident killer whales Orcinus orca of British Columbia. The site is also an active whale-watching area. A voluntary code of conduct requests that boats do not approach whales closer than 100 m to address perceived, rather than demonstrated, effects of boat traffic on killer whales. The purpose of the study was to test the relevance of this distance guideline. Relationships between boat traffic and whale behaviour were studied in 1995 and 1996 by shore-based theodolite tracking of 25 identifiable focal animals from the population of 209 whales. Individual killer whales were repeatedly tracked in the absence of boats and during approaches by a 5.2 m motorboat that paralleled each whale at 100 m. In addition, whales were tracked opportunistically, when no effort was made to manipulate boat traffic. Dive times, swim speeds, and surface-active behaviours such as breaching and spy-hopping were recorded. On average, male killer whales swam significantly faster than females. Whales responded to experimental approaches by adopting a less predictable path than observed during the preceding, no-boat period, although males and females used subtly different avoidance tactics. Females responded by swimming faster and increasing the angle between successive dives, whereas males maintained their speed and chose a smooth, but less direct, path. Canonical correlations between whale behaviour and vessel proximity are consistent with these conclusions, which suggest that weakening whale-watching guidelines, or not enforcing them, would result in higher levels of disturbance. High variability in whale behaviour underscores the importance of large sample size and extensive experimentation when assessing the impacts of human activity on killer whales.

This study considers the replies to a 14-item questionnaire, by 27 monozytogic (MZ) and 38 dizygotic (DZ) pairs. Another sample consisting of 48 sets of parents of twins (24 of whom were MZ and 24 DZ, not necessarily corresponding to the couples of twins actually studied) was used, to answer a questionnaire directly related the one put to the twin pairs. The results of statistical tests performed (canonical correlation and Fisher's discriminant) indicate that only in MZ twins does self-awareness outweigh pair-awareness. This does not seem to be related to any difference between MZ and DZ twins in the education/upbringing received from their parents.